If is a matrix with no negative real eigenvalues and all zero eigenvalues of are semisimple, the principal th root of can be computed by Newton's method or Halley's method, with a preprocessing procedure if necessary. We prove a new convergence result for Newton's method, and discover an interesting property of Newton's method and Halley's method in terms of series expansions. We explain how the convergence of Newton's method and Halley's method can be improved when the eigenvalues of are known or when is a singular matrix. We also prove new results on th roots of -matrices and -matrices, and consider the application of Newton's method and Halley's method to find the principal th roots of these special matrices.
